Evolution of non-thermal emission from shell associated with AGN jets

0.99 MB·English

by  H. Ito

draftversion January14,2011,accepted forpublication in the AstrophysicalJournal
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
EVOLUTION OF NON-THERMAL EMISSION FROM SHELL ASSOCIATED WITH AGN JETS
Hirotaka Ito1, Motoki Kino2, Nozomu Kawakatu3, and Shoichi Yamada4,5
draft version January 14, 2011, accepted for publication inthe Astrophysical Journal
ABSTRACT
We explore the evolution of the emissions by accelerated electrons in shocked shells driven by jets
in active galactic nuclei (AGNs). Focusing on powerful sources which host luminous quasars, we
1
evaluated the broadband emission spectra by properly taking into account adiabatic and radiative
1
cooling effects on the electron distribution. The synchrotron radiation and inverse Compton (IC)
0
scattering of various photons that are mainly produced in the accretion disc and dusty torus are
2
considered as radiation processes. We show that the resultant radiation is dominated by the IC

n emission for compact sources (. 10kpc), whereas the synchrotron radiation is more important for
a larger sources. We also compare the shell emissions with those expected from the lobe under the
J assumption that a fractions of the energy deposited in the shell and lobe carried by the non-thermal

3 electrons are ǫ 0.01 and ǫ 1, respectively. Then, we find that the shell emissions are
e e,lobe
1 brighter than the∼lobe ones at infra-r∼ed and optical bands when the source size is & 10kpc, and the
IC emissions from the shell at & 10 GeV can be observed with the absence of contamination from

] the lobe irrespective of the source size. In particular, it is predicted that, for most powerful nearby
E sources (L 1047 ergs s−1), TeV gamma-rays produced via the IC emissions can be detected by
j
H ∼ ∼
the modern Cherenkov telescopes such as MAGIC, HESS and VERITAS.
. Subject headings: particle acceleration — radiation mechanisms: non-thermal — galaxies: active —
h
galaxies: jets —
p
-
o
r 1. INTRODUCTION ber, the shocked shells are expected to offer site of par-
t
s Itiswellestablishedthatradio-loudactivegalacticnu- ticleaccelerationasinthe shocksofsupernovaremnants
a (SNRs) and therefore give rise non-thermal emissions
clei (AGNs) are accompanied by relativistic jets (e.g.,
[ (Fujita et al. 2007; Berezhko 2008). Hence, although
Begelman et al. 1984, for review). These jets dissipate
the observations at radio seems to be unsuccessful, non-
1 theirkineticenergyviainteractionswithsurroundingin-
thermal emissions from the shell may be accessible at
v terstellar medium (ISM) or intracluster medium (ICM),
higher frequencies. In fact, while not detected in radio,
3 andinflate a bubble composedofdeceleratedjetmatter,
4 whichisoftenreferredtoascocoon. Initially,thecocoon recentdeepX-rayobservationhavereportedthepresence
5 is highly overpressured against the ambient ISM/ICM of non-thermal emissions from the shell associated with
2 (Begelman & Cioffi 1989) and a strong shock is driven the radio galaxy Centaurus A (Croston et al. 2009).
1. into the ambient matter. Then a thin shell is formed Theoretically,emissionsbytheacceleratedparticlesre-
sidinginshockedshellshavebeenstudiedbyFujita et al.
0 around the cocoon by the compressed ambient medium.
(2007). However,they paidattentiononly to the the ex-
1 The thin shell structure persists until the cocoon pres-
tended sources of 100 kpc and the inverse Compton
1 suredecreasesandthepressureequilibriumiseventually ∼
(IC) scattering of external photons was not included in
: achieved (Reynolds et al. 2001).
v theradiativeprocesses. Thecoolingeffectsontheenergy
Whilelargenumberofradioobservationsidentifiedco-
i distribution of non-thermal electrons were not consid-
X coons with the extended radio lobe, no clear evidence
ered,either. Motivatedbythesebackgrounds,weexplore
of radio emissions is found for the shocked shells (e.g.,
r in this paper the temporal evolution of the non-thermal
a Carilli et al. 1988). Due to the lack of detections, in
emissions by the accelerated electrons in the shocked
the previous studies on the extragalactic radio sources,
shells, properly taking into account the Comptonization
it is usually assumed that non-thermal emissions are
ofphotonsofvariousoriginsaswellasthecoolingeffects
dominated by or originated only in the cocoon (e.g.,
on the electron distribution. Focusing on the powerful
Stawarz et al. 2008). However, since strong shocks are
sources which host luminous quasar in its core, we show
driven into tenuous ambient gas with a high Mach num-
that the shell can produce prominent emissions ranging
from radio up to 10 TeV gamma-ray and discuss the
[email protected]
∼
1 Department of Aerospace Engineering, Tohoku University, possibility for the detection.
6-6-01Aramaki-Aza-Aoba,Aoba-ku,Sendai,980-8579,Japan The paper is organizedas follows. In 2, we introduce
2NationalAstronomicalObservatoryofJapan,2-21-1Osawa, §
the dynamical model, which describes the evolution of
Mitaka,Tokyo181-8588,Japan
3Graduate School of Pure and Applied Sciences, Uni- the shell expansion, and explain how the energy distri-
versity of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8571 butionofelectronsresidingintheshellandthespectraof
[email protected] the radiations they produce are evaluated based on the
4Science and Engineering, Waseda University, 3-4-1 Okubo,
dynamical model. The obtained results are presented in
Shinjuku,Tokyo169-8555,Japan
5Advanced Research Institute for Science & Engineering, 3. In 4, we compare the emissions from the shell with
§ §
Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, those from the lobe and discuss its detectability. We
Japan
2 Ito et al.
close the paper with the summary in 5.
§
2. MODELFOREMISSIONSFROMTHESHELL
2.1. Dynamics
Inthissection,wegivethemodeltoapproximatelyde-
scribe the dynamics of the expanding cocoon and shell
and to provide a basis, on which the energy distribu- bow
tion of accelerated electrons and the radiation spectra shock
are estimated. The schematic picture of the model is
jet
illustrated in Fig. 1. For simplicity we neglect the elon-
gationofcocoonandshellinthejetdirectionandassume
that they are spherical. We also assume that the shell
cocoon shell
width, δR, is thin compared with the size of the whole
system, R,andignorethe difference betweenthe radiiof
the bowshockandthe contactsurface. This assumption contact
isvalidaslongastheexpansionvelocityhasahighMach surface
declining
number (see e.g., Ostriker & McKee 1988). We further
density
assume that the kinetic power of jet, L, is constant in
j
time on the time scale of relevance in this study.
Under the aboveassumptions,the dynamicsofthe ex-
pandingcocoonandshellcanbeapproximatelydescribed
bythemodelofstellarwindbubbles(Castor et al.1975). Fig.1.— The schematic picture of the model employed in this
The basic equations are the followings: (1) the equation study to approximate the expansion of the cocoon and shell pro-
ducedbythejetpushingintotheambientmediumfromAGN.
for the momentum of the swept-up matter in the shell
and (2) the equation for the energy in the cocoon. They
are expressed, respectively, as where γˆ is the specific heat ratio for the ambient
a
medium. By equating M with the mass in the shell
d s
M (t)R˙(t) =4πR(t)2P (t), (1) given by V ρ , where V = 4πR2δR is the volume of the
dt s c s s s
(cid:16) (cid:17) shell, the shell width is obtained as δR=(γˆ 1)[(γˆ +
a a
1)(3 α)]−1R . It is found that the ratio o−f δR to R
d Pc(t)Vc(t) dVc(t) does−not depend on time and the thin-shell approxima-
+P (t) =L, (2)
dt(cid:18) γˆc 1 (cid:19) c dt j tion is reasonably good for the typical values, γˆa = 5/3
− and 0 α 2.
where R˙ = dR/dt, Vc = 4πR3/3, and Pc are the expan- The≤tota≤l internal energy stored in the shell, Es =
sionvelocity of the shell, the volume andpressure of the P V /(γˆ 1), is one of the most important quantities,
s s a
cocoon, respectively, and γˆc is the specific heat ratio for since it de−termines the energy budget for the radiation.
the plasma inside the cocoon. The swept-upmass in the From Eqs. (3) and (4), E can be expressed as
s
shell is defined as M = R4πr2ρ (r)dr with the mass
s 0 a E =fLt, (5)
density of the ambient meRdium, ρa. s j
In this study, ρa is assumed to be given by ρa(r) = where f is a fraction of the total energy released by the
ρ0(r/r0)−α with r0 and ρ0 being the reference position jet(Ljt),whichisconvertedtothe internalenergyofthe
and the mass density there, respectively. Then Eqs. (1) shell, and is given by
and (2) can be solved analytically and the solution is
given as 18(γˆc 1)(5 α)
f = − − , (6)
(γˆ +1)2[2α2+(1 18γˆ )α+63γˆ 28]
L 1/(5−α) a − c c−
R(t)=CR r0α/(α−5) j t3/(5−α), (3) which turns out to be time-independent. For typical
(cid:18)ρ (cid:19)
0 numbers, γˆ = 4/3, γˆ = 5/3, 0 α 2, f depends
c a
≤ ≤
where C is a function of α and γˆ given as on α only weakly and f 0.1.
R c
∼
(3 α)(5 α)3(γˆc 1) 1/(5−α) 2.2. Energy Distribution of Electrons
C = − − − .
R (cid:20)4π{2α2+(1−18γˆc)α+63γˆc−28}(cid:21) Assuming that a fraction, ǫe, of the energy deposited
in the shell, E , goes to the non-thermal electrons, we
Employingthe non-relativisticRankine-Hugoniotcon- s
evaluate their energy distribution. We solve the kinetic
dition in the strong shock limit (Landau & Lifshitz
equationgoverningthe temporal evolutionof the energy
1959),whichisjustifiedbytheassumptionthattheshell
distribution of electrons, N(γ ,t), which is given as
expands with a high Mach number, we obtain from Eq. e
(3) the density, ρ , and pressure, P , in the shell, which
s s ∂N(γ ,t) ∂
e
are assumed to be uniform, as follows: = [γ˙ (γ )N(γ ,t)]+Q(γ ), (7)
cool e e e
∂t ∂γ
e
γˆ +1 R(t) −α
ρs(t)= γˆaa 1ρ0(cid:18) r0 (cid:19) , fwahcteorer,γteh,eγ˙ccooool(lγineg)=rat−edvγiea/addt,iaabnadtiQc(eγxep)aanrseiothnesLaonrdenratz-
−
2 R(t) −α diativelosses,andtheinjectionrateofnon-thermalelec-
P (t)= ρ R˙(t)2, (4) trons, respectively. The latter two, that is, the injection
s 0
γˆ +1 (cid:18) r (cid:19)
a 0
3
rate Q(γ ) and the cooling rate γ˙ , which will be de- the differential cross-section for IC scattering given in
e cool
scribed in detail below, are evaluated based on the dy- Blumenthal & Gould (1970).
namical model described in the previous section. The typical magnetic field strengths in el-
Following the theory of the diffusive shock accelera- liptical galaxies and clusters of galaxies are
tion(DSA) (Bell1978;Drury 1983;Blandford & Eichler a few µG (e.g., Moss & Shukurov 1996;
1987), we assume that the non-thermal electrons are in- Vikhlinin, Markevitch, & Murray 2001; Clarke et al.
jectedintothepost-shockregionwithapower-lawenergy 2001; Carilli & Taylor 2002; Schekochihin et al. 2005).
distribution of the form Assuming that as it passes through a shock wave, the
magnetic field is adiabatically compressed by a factor of
Q(γ )=Kγ−p for γ γ γ , (8)
e e min ≤ e ≤ max 1 4 depending on the obliqueness of the shock to the
∼
magnetic field lines, we choose B = 10µG as a fiducial
where γ and γ correspond to the minimum and
min max value for the magnetic field strength in the shell when
maximum Lorentz factors, respectively. In this study
evaluating the acceleration rate, γ˙ , and synchrotron
we set γ = 1 and the power-law index, p, is fixed accel
min cooling rate, γ˙ . The corresponding energy density of
to 2, an appropriate value for the linear diffusive shock syn
magnetic field is given by
acceleration in the strong shock limit. The maximum
Lraotree,nγ˙tz fa,cttoorthise oabcctealienreadtiobnyreaqtueagtiivnegntbhye the cooling UB ≈4.0×10−12B−25 erg cm−3, (13)
cool
3eBR˙2 whInereevBa−lu5a=tinBg/t1h0eµcGo.oling rate for IC scattering, γ˙ ,
γ˙ = , (9) IC
accel 20ξm c3 we take into account various seed photons of relevance
e
in this context. In their paper, Stawarz et al. (2008)
where B is the magnetic field strength in the shell. The explored high energy emissions by relativistic electrons
so-called “gyro-factor”, ξ, can be identified with the ra- in the radio lobes through the IC scatterings of various
tio of the energy in ordered magnetic fields to that in photons. Among themareUV emissionsfromthe accre-
turbulent ones (ξ = 1 is the Bohm limit). The nor- tiondisc,IRemissionsfromthedustytorus,stellaremis-
malisation factor, K, in the injection rate is determined sionsinNIRfromthehostgalaxyandsynchrotronemis-
from the assumption that a fraction, ǫe, of the shock- sionsfromtheradiolobe. Inadditiontotheseemissions,
dissipatedenergyiscarriedbythenon-thermalelectrons: we also consider CMB as seed photons in the present
γγmmianx(γe − 1)mec2Q(γe)dγe = ǫedEs/dt = fǫeLj. A study. Following Stawarz et al. (2008), we assume that
Rrough estimation of the normalizationfactor is obtained the photons from the disc, torus, host galaxy, and CMB
as K 0.1ǫ L/[m c2ln(γ )], where we used f 0.1 are monochromatic and have the following single fre-
e j e max
( 2.1).∼It is noted that since the factor K is propor∼tion- quencies: ν = 2.4 1015 Hz, ν = 1.0 1013 Hz,
UV IR
a§tetoǫ andL,theresultantluminosityofnon-thermal ν = 1.0 1014 Hz,×and ν = 1.6 101×1 Hz. The
e j NIR CMB
× ×
emissionsalsoscalesinthesamemannerwiththesequan- photons from the radio lobe are assumed to have a con-
tities. tinuous spectrum given by L ν−0.75.
ν,lobe
∝
In the cooling rate, γ˙ , both radiativeand adiabatic Fromthe luminosity, L , ofthe UV emissionsby the
cool UV
losses are taken into account. As for the former, the accretiondisc,theenergydensityofthesephotonsinthe
synchrotron radiation and the IC scattering off photons shell is given approximately as
of various origins. The cooling rate for the adiabatic
L
dynamical expansion is expressed for electrons with a U = UV
Lorentz factor γ as UV 4πR2c
e
3.0 10−9L R−2 erg cm−3, (14)
1R˙ ≈ × UV,46 1
γ˙ad = γe. (10) where L = L /1046erg s−1, and R = R/1kpc.
3R UV,46 UV 1
As is assumed in Stawarz et al. (2008), we take L =
UV
The rates of the radiative coolings via the synchrotron 1046 ergs s−1 for sources with jet kinetic power of L >
j
radiation and IC emissions are given, respectively, as 1045 ergs s−1, and L = 1045 ergs s−1 for L
UV j
4σ U 1045 ergs s−1 as fiducial values. The adopted values ar≤e
γ˙syn = 3mT cBγe2, (11) appropriate for sources hosting a luminous quasar. Ac-
e
cording to the broadband observations of quasars (e.g.,
and Elvis et al.1994;Jiang et al.2006),theIRandUVemis-
sions are approximately comparable although there are
4σ U
γ˙ = T phγ2F (γ ), (12) some variations in the relative strengths from source to
IC 3mec e KN e source. We hence assume that the luminosity of the IR
emissionsfromthe torus,L ,isequalto thatoftheUV
where σ , c and m are the cross section of Thomson IR
T e
emissions from the disc, L , and the energy density of
scattering, the speed of light and the electron mass, re- UV
IR photons in the former case is estimated as
spectively. The energy densities of magnetic fields and
pUhpoht,ornesspinecttihveelysh.elTlhaerefudnecntoiotendFbKyN(UγBe)=enBco2d/e8sπbaontdh UIR = 4πLRIR2c ≈3.0×10−9LIR,46R1−2 erg cm−3,(15)
the distributions of seed photons and the Klein-Nishina
(KN) cross section for the Compton scattering and re- where L = L /1046 erg s−1. In evaluating the en-
IR,46 IR
duces to unity in the Thompson limit. Note, however, ergydensity ofphotonsfromthe hostgalaxy,we assume
we do not employ this limit but calculate F (γ ) from that most of the photons are emitted by stars in the the
KN e
4 Ito et al.
core region with the radius of 1 kpc (de Ruiter et al. 2.3. Radiation spectra
∼
2005). Considering only the sources with R & kpc, we
From the energy distribution of non-thermal electrons
can estimate the energy density of the optical photons
justobtained,the spectrumofthe synchrotronradiation
from the luminosity, L , as
NIR is calculated as
L γmax
V
UNIR=4πR2c Lν,syn =Z Psyn(ν,γe)N(γe)dγe. (20)
3.0 10−10L R−2 erg cm−3, (16) γmin
≈ × NIR,45 1 Here, P (ν,γ ) is the pitch-angle-averagedpowerspec-
syn e
where L =L /1045ergs s−1. The energydensity trum for a single electron given by
NIR,45 NIR
of CMB photons is given by
√3e3B π ν
P (ν,γ )= sin2θF dθ,
U 4.2 10−13 erg cm−3. (17) syn e 2m c2 Z (cid:18)ν (cid:19)
CMB ≈ × e 0 c
Theredshiftisignoredforsimplicity. Finally,theenergy where θ is the pitch angle and ν = 3eBγ2sinθ/4πm c
c e e
density of photons emitted from the lobe is obtained by is the characteristic frequency of the emitted photons.
∞
assumingthatafractionη ofthe jetpowerisradiatedas ThefunctionF(x)isdefinedbyF(x)=x K (y)dy,
x 5/3
radio emissions from the lobe (i.e., Lν,lobedν = ηLj). where K5/3(y) is the modified BesselfuncRtion of the 5/3
Assuming η =10−2 as a fiducial caseR, it is given as order (Rybicki & Lightman 1979). Note that the syn-
chrotron self-absorption is ignored, since it is important
Lν,lobedν only at low frequencies below 107Hz in the current con-
U =
lobe R 4πR2c text.
≈3.0×10−12η−2L45R1−2 erg cm−3, (18) obItnaitnheedsaasme way, the spectrum of the IC scattering is
where η−2 =η/10−2 and L45 =Lj/1045 ergs s−1. γmax
Thecoolingrateforthe IC scatteringisobtainedfrom L = P (ν,γ )N(γ )dγ , (21)
ν,IC IC e e e
Eq. (12) by plugging in the energy densities Uph ob- Zγmin
tained above and calculating F (γ ) for the assumed
KN e where
photon distributions. Each contribution from the disc,
∞
torus, host galaxy, CMB, and lobe is denoted as γ˙ ,
IC,UV P (ν,γ )=hνc n (ν )σ (ν,ν ,γ )dν .
γ˙IC,IR, γ˙IC,NIR, γ˙IC,CMB, and γ˙IC,lobe. The total cooling IC e Z0 ph s IC s e s
rate γ˙ is the sum of the rates for the adiabatic, syn-
cool
is the power spectrum for a single electron in isotropic
chrotron and IC losses. (γ˙ = γ˙ +γ˙ +γ˙ +
cool ad syn IC,UV
photon fields. Here, n (ν ) and σ (ν,ν ,γ ) are the
γ˙ +γ˙ +γ˙ ). Note that the synchrotron ph s IC s e
IC,NIR IC,CMB IC,lobe number density of seed photons per unit frequency and
self-Compton is ignored, since its effect is negligible in
thedifferentialcrosssectionfortheICscatteringgivenby
any case considered in the present study.
Blumenthal & Gould (1970), which is valid both in the
Now that the injection rate, Q(γ ), and the cooling
e ThompsonandKNregimes. Theoriginsoftheseedpho-
rate, γ˙ , have been evaluated, the energy distribution
cool
tons have been discussed in 2.2. The number densities
ofnon-thermalelectrons,N(γ ,t),isobtainedbyputting
e §
perunitfrequencyofthesephotonsarethengivenasfol-
these quantities in Eq. (7). Although Q(γ ) is time-
e lows: n (ν )=(U /hν )δ(ν ν ) for the UV emis-
dependent through γ and so is γ˙ because of the ph s ph ph s ph
max cool −
sions from the accretion disc (U = U ,ν = ν ),
power-dependenceoftheexpansionrate(seeEq.(3)),we ph UV ph UV
the IR emissions from the torus (U = U ,ν =
ignorethesevariationsoverthedynamicaltimescale t. ph IR ph
∼ ν ), the NIR emissions from the host galaxy (U =
Then, employing the instantaneous values of γ˙ (γ ) IR ph
cool e
U ,ν = ν ), and CMB (U = U ,ν = ν );
andQ(γ )evaluatedateachtimeandfixingthem,wecan V ph V ph CMB ph CMB
solve Eqe. (7) by the Laplace transforms (Melrose 1980; nph(νs)=Lν,lobe/hνs4πR2c for the lobe emissions.
Manolakouet al. 2007) and the solution can be written
as 3. BROADBANDSPECTRUMOFTHESHELL
In the following we focus on powerful radio sources
N(γe,t)= γ˙coo1l(γe)Zγeγudγe′Q(γe′), (19) (pLrejs∼en1t0t4h5e−t1e0m47peorrgaslse−v1o)luhtoiostnisngoflutmheineonuesrqgyuadsaisrtsriabnud-
tionofnon-thermalelectronsandtheresultantradiation
where the upper limit of the integral is given by
spectra.
As a fiducial case, we set the parameters for the am-
γu =(cid:26)γγ∗m(aγxe) ffoorr γγmbrin<≤γeγe≤≤γmγbarx,. αbie=nt1.m5,ait.et.e,rρa(sr)ρ=0 0=.1m0.1ρmp(rc/m1−k3p,cr)0−1=.5cm1−k3p,cwhaenrde
a p 0.1
Here γ∗ and γbr are determined by the relation mp is the proton mass and ρ0.1 = ρ0/0.1mp. These
τ(γ∗,γe) = τ(γmax,γbr) = t, where the function are indeed typical values for elliptical galaxies (e.g.,
′ γ′ ′′ ′′ Mulchaey & Zabludoff1998;Mathews & Brighenti2003;
τ(γ ,γ) = dγ /γ˙ (γ ) gives the time it takes an
γ cool Fukazawa, Makishima, & Ohashi 2004; Fukazawa et al.
electron toRcool from γ′ to γ in the Lorentz factor. Note 2006). We adopt γˆ = 4/3 and γˆ = 5/3. Then, R is
c a
thatγbr correspondstotheLorentzfactor,wheretheen- given from Eq. (3) as
ergydistributionofnon-thermalelectronsshowsabreak
−2/7 2/7 6/7
owing to the radiative coolings (see 3.1 for detail). R(t) 22ρ L t kpc, (22)
§ ≈ 0.1 45 7
5
where t = t/107 yr. As mentioned in 2.2, the power- γ R−7/6 in the synchrotron-dominated stage, since
7 br
law index, p, for the injected non-therm§al electrons (see the ∝relation t γ−1U−1 γ−1R0 holds. Regard-
Eq. (8)) is fixed to 2. Although the parameters ξ and ing the depencdoeonlc∝e onetheBjet∝poweer, for a given source
ǫe which characterize the electron acceleration efficien- size R, γ increases with L roughly as γ L1/3 be-
cies are highly uncertain, it is natural to expect that br j br ∝ j
cause the dependencies of the source age and the radia-
the range of these values are similar to those observed
tivecoolingtime scaleonthe jetpowercanbe expressed
in the SNRs, since the nature of shocks considered here
resemblesthoseintheSNRsinthattheyarestrongnon- as t∝Lj−1/3 and γbr ∝γe−1L0j, respectively.
relativistic collision-less shocks driven into ISM. Here Among the contributions to the IC losses, the scat-
we take ξ = 1, which corresponds to the Bohm diffu- tering of the IR torus photons is the largest at least in
sion limit, and ǫ = 0.01 as illustrative values, based the IC-dominated stage thanks to the high energy den-
e
on the the observations of SNRs (e.g., Dyer et al. 2001; sity of the IR photons. Although the UV disc photons
Ellison et al. 2001; Bamba et al. 2003; Yamazaki et al. are assumed to have the same energy density as the IR
2004; Stage et al. 2006; Tanaka et al. 2008). The lumi- photons (U = U ), the cooling by the former is sup-
UV IR
nosities of the emissions from accretion disc and dust pressed for γ & m c2/4hν 1.3 104 by the KN
e e UV
torus are taken as L = L = 1046 ergs s−1 for effect as seen in Figs. 2 and 3.∼As a r×esult, unless L
UV IR UV
L > 1045 ergs s−1, and L = L = 1045 ergs s−1 exceeds L significantly, most electrons with γ & 104
j UV IR IR e
for L 1045 ergs s−1, while fixed value of L = coolpredominantlythroughthe ICscatteringofIRpho-
j NIR
1045ergs≤s−1 is adopted for the emissions from the host tons. Then the transition from the IC-dominated stage
galaxy. The parameters for the magnetic field and lobe to the synchrotron-dominatedstage occurs roughly at
emissions are chosen as B = 10µG and η = 0.01. The R 27L1/2 B−2kpc, (23)
employed values of the parameters are summarized in IC/syn ∼ IR,46 −5
Table 1. which correspondsto the condition U U . Since the
IR B
∼
IC scattering of IR torus photons is also suppressed by
3.1. Evolution of Non-thermal Electrons
the KN effect above the Lorentz factor given by
In Figs. 2 and 3, we display the cooling and ac-
celeration time scales (left panels) and the energy dis- γKN =mec2/4hνIR 3 106,
∼ ×
tribution of non-thermal electrons (right panels). The the coolingsby the synchrotronemissions and/orthe IC
jet powers are chosen to be Lj = 1045 ergs s−1 and scattering of lobe photons are important at the high en-
L = 1047 ergs s−1 in Figs. 2 and 3, respectively. The ergyendofnon-thermalelectrons(γ .γ .γ )for
j KN e max
top,middleandbottompanelsinthefiguresaregivenfor the source with R < R (see, e.g., the middle left
IC/syn
the source sizes of R = 1, 10, and 100 kpc, respectively, panelofFig. 2andthetopandmiddleleftpanelsofFig.
which in turn corresponds to the different source ages. 3). For even larger sources with R > R , the syn-
IC/syn
Inadditiontothetotalcoolingtime scale,t ,weshow chrotronemissions are the dominant cooling mechanism
cool
the contributions from the adiabatic loss, t = γ /γ˙ , atallenergies. ThecontributionsfromtheICscatterings
ad e ad
synchrotron radiation, t = γ /γ˙ , and IC scatter- ofCMBandhostgalaxyphotons,ontheotherhand,are
syn e syn
ings of the UV disc photons, t = γ /γ˙ , IR modest at most during the whole evolution.
IC,UV e IC,UV
torus photons, t =γ /γ˙ , NIR host galaxy pho- ThemaximumLorentzfactoroftheinjectedelectrons,
IC,IR e IC,IR
tons, t = γ /γ˙ , CMB photons, t = which is determined by the condition t = t (or
IC,NIR e IC,NIR IC,CMB cool accel
γe/γ˙IC,CMB, and lobe photons, tIC,lobe = γe/γ˙IC,lobe, to- equivalently γ˙cool = γ˙accel), is γmax 107 109 for
gether with the source age, t. the source sizes R 1 100 kpc an∼d the −jet power
As mentioned in 2.2, the Lorentz factor at the spec- L 1045 1047 er∼gs s−−1. Regarding the dependence
tralbreak,γbr,corre§spondstotheelectronenergy,above onj ∼the sour−ce size, γmax increases with R for compact
which cooling effects become important. It is hence de- sourcesinwhichthecoolingatγ γ isdominatedby
e max
termined roughly by the condition t tcool as shown the IC emissions. This is because∼the cooling time scale
in the left panels of Figs. 2 and 3. Sin∼ce the adiabatic increases rapidly with size (t γ−1F (γ )−1U−1
loss is dominant over the radiative coolings only below γ−1F (γ )−1R2), in contracsotowl ∝ithetheKslNoweincreapshe i∝n
γ , the latter is always more important when the cool- e KN e
br acceleration time scale (t γ R˙−2 γ R1/3). On
ings affect the electron distribution. The relative im- accel e e
∝ ∝
the other hand, for larger sources in which the cool-
portance of the synchrotron emissions and IC emissions
ing at γ γ is dominated by the synchrotron
dependsontheelectronenergyaswellastheenergyden- e max
∼
emissions, γ decreases slowly with the source size
sitiesofmagneticfieldsandphotons. Whenthesourceis max
young and hence small, the energy loss is dominated by (γmax R−1/6), since the cooling time scale is indepen-
the IC emissions, since the energy density of photons is dent o∝f the size (t γ−1U−1 γ−1R0). Note that
cool ∝ e B ∝ e
largerthanthatofmagneticfields(referredtoastheIC- thistransitiontakesplaceearlierthanthetransitionfrom
dominated stage). As the the source becomes larger, on the IC-dominated stage to the synchrotron-dominated
the other hand, the energy density of photons decreases stage mentioned above owing to the suppression of IC
(U R−2)andthesynchrotronlossbecomesmoreim- emissionsinthe highenergyelectrons(γ &γ )by the
ph e KN
∝
portant(thesynchrotron-dominatedstage). Intheinitial KN effect. For a given source size, higher values of γ
max
IC-dominated stage, the break Lorentz factor increases areobtainedforsourceswithlargerjetpowers,since the
rweiltahtiothnes sto∝urcRe7s/i6zeanads γtcborol∝∝Rγ5e/−61,Uwp−hh1er∝e wγe−e1uRse2d. tOhne acTcehleerraetsiuonltatnimteensecraglyedisissthroibrutetrio(ntsacocfeln∝onγ-ethLej−r2m/3a)l.elec-
the other hand, γ decreases with the source size as trons, N(γ ), are understood as follows. They have a
br e
6 Ito et al.
TABLE 1
List of the ModelParameters.
Parameters Symbols EmployedValues
power-lawindexfordensityprofileofambientmatter,ρa(r) α 1.5
referencepositioninρa(r) r0 1kpc
massdensity,ρa(r0),atr=r0 ρ0 0.1mp cm−3
magneticfieldstrength B 10µG
luminosityofUVemissionsfromaccretiondisc LUV 1046 ergss−1 (forLj>1045 ergss−1)
1045 ergss−1 (forLj≤1045 ergss−1)
luminosityofIRemissionsfromdusttorus LIR 1046 ergss−1 (forLj>1045 ergss−1)
1045 ergss−1 (forLj≤1045 ergss−1)
luminosityofNIRemissionsfromhostgalaxy LNIR 1045 ergss−1
ratiooflobeluminositytojetpower η 0.01
power-lawindexforenergydistributionofinjectedelectrons p 2
gyro-factor ξ 1
energyfractionofnon-thermalelectrons ǫe 0.01
sharp cut-off at γ = γ whereas the original spec- sources. This feature is not so remarkable for most of
e max
trum N(γ ) γ−2 of the injected electrons is retained the IC components mainly because the emissions them-
e ∝ e
at low energies below γ . Since the cooling effects selvesaresuppressedbytheKNeffect. TheICscattering
br
are negligible, the energy spectrum for γ . γ are of IR dust-torus photons dominates over other IC com-
e br
roughly given by N(γe) Q(γe)t. The spectrum steep- ponents up to the source size of R 85L1/2 kpc. For
ens for γ & γ owing∼to the radiative cooling and is ∼ IR,46
e br larger sources, on the other hand, the IC scattering of
roughlydetermined by the energy dependence of t as
cool CMB photons becomes more important, since CMB has
N(γ ) Q(γ )t (γ ). Asmentionedabove,forsources
e ∼ e cool e the largest energy density of all photons considered in
smaller than R , the IC scattering of IR torus pho-
IC/syn this paper. The contributions from the UV disc photons
tons determines the spectral shape up to γ , where
∼ KN andNIRhostgalaxyphotonsaremodestatbestthrough
theKNeffectkicksinandhardenstheenergyspectrafor
the entire evolution.
higher energies (see, e.g., the top and middle panels of
The energy injection rate above γ is roughly equal
Figs. 2 and 3). This feature is absent for larger sources br
to the radiative output (γ2m c2Q(γ ) νL ) of non-
(R>R ), because the synchrotron loss (γ˙ γ2) e e e ∼ ν
IC/syn syn ∝ e thermalelectronsinthisenergyregimebecausethecool-
is dominant at all energies and the power is simply re-
ing time scale is shorter than the dynamical time scale.
duced by 1 (N(γ ) γ−3) from the original spectrum
e ∝ e Sincetheenergyinjectionrateisindependentoftheelec-
of the injected electrons. For given source size, higher tronenergy(γ2Q(γ ) γ0),these non-thermalelectrons
values of N(γ ) are obtained for sources with larger L, e e ∝ e
e j produce a rather flat and broad spectrum (νL ν0)
since the electron injection rate Q(γe) is higher. in the correspondingfrequency range. Fromtheνre∝lation
γ2m c2Q(γ ) = Km c2 0.1ǫ L/[ln(γ )] (see 2.2),
3.2. Evolution of Radiation Spectra e e e e ∼ e j max §
we can give a rough estimate to the peak luminosity as
In Fig. 4 we show the radiation spectra, νL , which
are obtained by inserting into Eqs. (20) and ν(21) the (νLν)peak 4.0 1040ǫ−2L45 ergs s−1, (24)
∼ ×
energy distributions of non-thermal electrons in the pre- whereǫ−2 =ǫe/0.01. Intheaboveequations,weignored
vious section. It is assumed that the jet powers are the dependence of luminosity on γ because the lu-
Lj = 1045 ergs s−1 (left panels) and Lj = 1047 ergs s−1 minosity only scale logarithmicallymwaixth its value, and
(right panels). Thetop,middleandbottompanelsofthe employed a typical value γ 108. The feature is
max
figurecorrespondtothesourcesizesofR=1kpc,10kpc clearly seen in Fig. 4. Indeed, th∼e spectra are flat with
and 100 kpc, respectively. In addition to the total lumi- a peak luminosity given approximately by Eq. (24). As
nosity (thick solid line), we show the contributions from mentionedin 2.2, the emissionluminosity scale approx-
the synchrotron emissions (thin solid line) and the IC imately linear§ly with the acceleration efficiency ǫ and
e
scatterings of UV disc photons (long-short-dashed line), the jet power L. For given values of ǫ and L , while
j e j
IRtorusphotons(dot-dashedline),NIRhostgalaxypho- the value of (νL ) remains nearly constant, the fre-
ν peak
tons (dotted line), CMB photons (long-dashed line) and quencyrange,wherethespectrumisflat,varieswiththe
lobe photons (short-dashed line). source size because of the changes in γ , γ and the
br max
Thesynchrotronemissionsarethe mainlow-frequency main emission mechanism (synchrotron or IC). It is em-
component, which extends from radio to X-ray phasized that the peak luminosity is chiefly governedby
2(γmax/108)2B−5 keV. The IC emissions become re∼- ǫe and Lj and is quite insensitive to the magnetic field
markable at higher frequencies up to gamma-ray strengthandseedphotons,whichwillonlyaffectthefre-
50(γmax/108) TeV. As mentioned in 3.1, the dominan∼t quency range of the flat spectrum. This means that if
§
radiative process changes from the IC emissions to the L is constrained by other independent methods (e.g.,
j
synchrotronradiations as the source becomes larger. As Allen et al. 2006; Ito et al. 2008), the observation of the
a result, the former is more luminous than the latter peak luminosity will enable us to obtain information on
whenthesourceisyoungandcompact(R.RIC/syn)and the highly unknown acceleration efficiency ǫe.
viceversa. Owningtothehardeningoftheenergydistri-
butions of non-thermal electrons at γ & γ (see Figs. 4. COMPARISONWITH EMISSIONSFROMTHELOBE
e KN
2 and 3) for compact sources (R . R ), their syn- Despite our focus on the non-thermal emission from
IC/syn
chrotron spectrum becomes also harder than for larger the shell, emission from the lobe (or, cocoon) is an an-
7
Fig.2.— Cooling and acceleration time scales (left panels) and energy distributions of non-thermal electrons (right panels) for sources
with the jet power of Lj = 1045ergss−1. The top, middle and bottom panels are shown for the source sizes of R = 1, 10 and 100 kpc,
respectively. Thethick linesinthe leftpanels givethetotal coolingtimescale(thick solid line), the acceleration timescale(thick dashed
line)andthesourceage(thickdot-dashedline)whereasthethinlinesarecontributionstothetotalcoolingtimescalefromvariousprocesses:
adiabatic losses (dot-dot-dashed line), synchrotron emissions (thin solid line), and IC scatterings of UV disc photons (long-short-dashed
line),IRtorusphotons(thindot-dashedline),NIRhostgalaxyphotons(dottedline),lobephotons(thinshort-dashed line)andCMB(thin
long-dashed line)photons.
otherimportantingredientwhicharisesasaconsequence comparisonbetween the spectra of the lobe and shell.
of interaction of jet with ambient medium. As shown
by Stawarz et al. (2008), lobes can produce very bright 4.1. Model for Emissions from the Lobe
emissions in broadband. Therefore, in order to consider We basically follow the same procedure employed for
the application of our model to the observation of radio theshellincalculatingtheenergydistributionoftheelec-
sources, it is essential to investigate the relative signif- tronsandthe resultingemissions. Since weareconsider-
icance of the two emissions, since the lobe component ing the electrons residing in the lobe, the energy stored
may hamper the detection. In this section, we evaluate in the cocoon, E =P V /(γ 1), is the energy budget
c c c c
the emissions from the lobe and show the quantitative for the emissions. From the d−ynamical model described
8 Ito et al.
Fig.3.—SameasFig. 2butforthejetpowerofLj=1047 ergss−1.
in 2.1, the total internal energy stored in the cocoon is of Eq. (7). In the present study, we assume that the
§
given by electrons injected into the lobe have a power-law energy
distribution given as
E =f Lt, (25)
c lobe j
Q(γ )=K γ−plobefor γ γ γ .
wheref =(5 α)(7 2α)/[2α2+(1 18γˆ )α+63γˆ e lobe e min,lobe≤ e ≤ max,lobe
lobe c c
28]. For γˆ =4/3−, γˆ =−5/3 and α=1.−5, f =7/13− While the employed values of the power-law index and
c a lobe
0.5 is obtained. As can be seen from the above equatio∼n minimum Lorentz factor are the same as those adopted
and Eq. (5), Ec is larger than Es by a factor of 5. for the shell (plobe = 2 and γmin,lobe = 1), the max-
The energy distribution of the electrons in the∼lobe is imum Lorentz factor is fixed as γ = 105. We
max,lobe
determined based on Eq. (7) as in the case of the shell. will comment on the assumed value of γ later
max,lobe
Asdescribedin 2.2,byevaluatingtheelectroninjection in 4.2. The normalisation factor, K , is deter-
lobe
§ §
rate, Q(γ ), and the cooling rate, γ˙ (γ ), the energy mined from the assumption that a fraction, ǫ , of
e cool e e,lobe
distributionofthe electronsis obtainedby putting these the energy deposited in the lobe is carried by non-
quantities in Eq. (19) which corresponds to the solution thermal electrons: γmax,lobe(γ 1)m c2Q(γ )dγ =
γmin,lobe e − e e e
R
9
Fig.4.— Broadband emission produced within the shell of sources with the jet powers of Lj = 1045ergss−1 (left panels) and Lj =
1047ergss−1 (right panels). Thetop, middleandbottom panelsaredisplayedforthesourcesizesofR=1,10,and100kpc,respectively.
The various lines show the contributions from the synchrotron emissions (thin solid line) and IC scatterings of UV disc photons (long-
short-dashed line), IR torus photons (dot-dashed line), NIR host galaxy photons (dotted line), CMB photons (long-dashed line) and lobe
photons (short-dashed line). Thethicksolidlineisthesumoftheseemissions.
ǫ dE /dt = f ǫ L. A rough estimation of of magnetic field is given as
e c lobe e,lobe j
the normalization factor is obtained as K
00..55ǫ.e,loRbeegLaj/rd[minegc2tlhne(γmcoaox,llionbge)]r,atwehoerfethwee eulseecdtrolfonlboseb,ew∼∼e UB = ǫBVEcc ≈4.0×10−9ǫB,0.1ρ10/.13L245/3R1−11/6erg cm−3,
take into account the adiabatic losses as well as radia-
where ǫ = ǫ /0.1. As for the IC cooling rate, we
B,0.1 B
tivelossesduetothesynchrotronandICemissions. The
take into account all seed photon fields which are con-
adiabatic cooling rate is evaluated from Eq. (10) and
sidered in the shell except for the synchrotron photons
coincides with that of the shell. The synchrotron cool-
from the lobe. Considering the IC of lobe photons, al-
ing rate is determined from Eq. (11) under the assump-
though a particular spectrum was assumed in the case
tion that a fraction ǫ , of the energy E is carried by
B c of the shell (see 2.2), to be self-consistent, the distri-
the magnetic fields. The corresponding energy density §
bution of the seed photons should be determined from
10 Ito et al.
the calculated synchrotron emissions (synchrotron self- is larger than that of the shell. As in the case of the
Compton). However, the method given in 2.2 used to shell, the luminosity scales linearly with ǫ and L
e,lobe j
§
solveEq. (7)cannotbeappliedwhenaseedphotonfield (see 2.3). A rough estimate to the peak luminosity is
§
has dependence on the electron distribution. Therefore, obtained as
since the cooling rate due the synchrotronself-Compton
is modest at best in any case considered in the present (νLν)peak,lobe 2.0 1043ǫe,lobeL45 ergs s−1. (26)
∼ ×
study,we neglectedits effect merelyforsimplicity. As in
It should be noted that the photon fluxes displayed in
the case of the shell, we assume that the photons from
the figure correspond to an upper limit since ǫ = 1
thedisc,torus,hostgalaxyandCMBaremonochromatic e,lobe
is assumed.
with frequencies givenin 2.2. On the other hand, while
§ From Eqs. (24) and (26), the ratio between the peak
same value is employed for CMB, the energy densities
luminosities of the emissions from the lobe and shell is
of the disc, torus and host galaxy photons are taken to
obtained as 5ǫ /ǫ . Hence, the contrast between
be larger than those in the shell by a factor of 3, since ∼ e,lobe e
thetwoemissionsisdeterminedbytheparametersǫ
the lobe is located closer to the emitting sources. The e,lobe
andǫ . If ǫ is much largerthan ǫ , the overallspec-
cooling rate due the IC of these photons are evaluated e e,lobe e
tra is dominated by the lobe. Although there is little
from Eq. (12) based on the above mentioned spectrum
constraint on the values of ǫ and ǫ , it is expected
and energy densities. e,lobe e
that the ratio ǫ /ǫ is indeed large in most of the
Theemissionspectrumofthe lobeis calculatedbyfol- e,lobe e
radio sources as is examined here due to the fact that,
lowing the procedure described in 2.3. From the ob-
§ while large number of radio observations identified non-
tained energy distribution of the non-thermal electrons,
thermal emissions from the lobe, no clear evidence of
spectraofthesynchrotronandICemissionareevaluated
the shell emissions have been reported so far at radio.
fromEqs. (20)and(21),respectively. Themagneticfield
It is emphasized, however, that, even in the case where
and seed photon fields considered above is used for the
ǫ is significantly larger than ǫ , there are frequency
calculation. In addition, although we did not consider e,lobe e
ranges in which the emissions from the shell can over-
its contributiononthe cooling rate,we also evaluate the
whelm those from the lobe as is seen in Fig. 5. This is
IC of the photons from the lobe based on the calculated
duetothedifferencebetweenthemaximumenergyofthe
synchrotronspectrum.
electrons in the lobe and shell. Since γ 108 largely
max
4.2. Lobe vs Shell exceedγ =105,thesynchrotronand∼ICemissions
max,lobe
fromtheshellextenduptomuchhigherfrequenciesthan
For an illustrative purpose,here we examine the emis-
those from the lobe, and, as a result, the shell can dom-
sions from the lobe by assuming ǫ = 1, an extreme
e,lobe
inate the emission spectra at the frequencies above the
case where all energy stored in the cocoon is converted
cut-off frequencies of the synchrotron and IC emissions
to that of the non-thermal electrons, and ǫ = 0.1, a
B
from the lobe.
magnetic field strength factor of few below the equipar-
Regardingthesynchrotronemission,althoughthehigh
tition value. These values are identical to those em-
frequencyendofthe emissionisoverwhelmedby the low
ployed in Stawarz et al. (2008). In Fig. 5 we show
frequency tail of the IC emissions from the lobe, emis-
the obtained photon fluxes, νF , from the lobe. In or-
ν
sions from the shell can become dominant at frequen-
der to illustrate the comparison between the emissions
cies above the cut-off frequency of synchrotron emis-
from the lobe and the shell, photon fluxes from the shell
sion from the lobe which is roughly given as hν
are also shown in the figure. Same set of parameters
∼
which was previously assumed (Table. 1) is employed 6×10−2(γmax,lobe/105)2ǫ1B/,02.1ρ10/.16L145/3R−11/12 eV when
for the emissions from the shell. It is assumed that the source size is large (R & R ). For example,
the jet powers are Lj = 1045 ergs s−1 (left panels) and emissions from the shell can be oIbCs/esryvned at frequencies
L = 1047 ergs s−1 (right panels) and the source is lo- from IR to optical without strong contamination from
j
cated at a distance of D = 100 Mpc. The top, mid- the lobe for sources with size of R 100 kpc (see the
∼
dle and bottom panels of the figure correspond to the bottom panels of Fig. 5). Hence, deep observations of
source sizes of R = 1 kpc, 10 kpc and 100 kpc, re- radiogalaxiesatthese frequencies maylead to discovery
spectively. We also plot the sensitivities of the Fermi of the shell emissions. It is noted that a broader range
satellite (http://www-glast.stanford.edu/) and HESS of frequencies could be detectable if the ratio ǫ /ǫ
e,lobe e
(http://www.mpi-hd.mpg.de/hfm/HESS/) and MAGIC is smaller. For compact sources(R.R ), however,
IC/syn
(http://magic.mppmu.mpg.de/) telescopes. The dashed synchrotron emissions are likely to be completely over-
and solid lines display the photon fluxes from the lobe whelmed by the lobe components, since the synchrotron
and shell, respectively. In addition to the total photon emissions from the shell is strongly suppressed (see the
flux (thick black line), the contributions from the syn- top panels of Fig. 5).
chrotron emissions (thin black line) and the IC scatter- Ontheotherhand,ICemissionsfromtheshellarenot
ingsofUVdiscphotons(blueline),IRtorusphotons(red subject to the contamination from the lobe irrespective
line), NIR host galaxy photons (light blue line), CMB to the size of the source at frequencies above the cut-off
photons (green line) and lobe synchrotronphotons (pur- frequencyofICemissionsfromthe lobewhichisroughly
ple line) are also shown. given as hν γ m c2 50(γ /105) GeV,
max,lobe e max,lobe
∼ ∼
As is seen in the figure, IC emissions from the lobe since the lobe emissions cannot extend above the fre-
tend to be brighter when the source size is smaller as in quency. It is also noted that compact sources (R .
the case of the shell. On the other hand, however, the R ) are favored for detection, since the luminosity
IC/syn
emissions are synchrotron-dominated even for compact ishigher. Contrarytotheaboveargument,althoughthe
sources(R 1 10kpc)sincethemagneticfieldstrength same value was assumed for the maximum Lorentz fac-
∼ −